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CHAPTER II.

FUNDAMENTAL OPERATIONS.

ADDITION.

30. Addition is the operation of finding the simplest equivalent expression for the aggregate of two or more algebraic quantities. Such expression is called their sum.

31. When the Terms are Similar and have Like Signs.

+

(1) What is the sum of a, 2a, 3a, and 4a ?

Take the sum of the coefficients, and annex the common unit or literal part. The first term (a) has a coefficient 1 understood ( 13).

a + 2a + 3a + 4 a

+ 10α

(2) What is the sum of 2 ab, 3 ab, 6 ab, and ab ?

NOTE. — When no sign is written, the sign + is understood (8 5).

2 ab 3 ab 6 ab ab

12 ab

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9 abd 15 abd

7 ab

3 abc

- 2 ad

15 ab

+ 7 abc

5 ad

- 24 abd

Hence, when the terms are similar and have like signs,

Add the coefficients, and to their sum prefix the common sign. To this annex the common unit or literal part.

Exercises.

1. 9 ab + AX

3 ab + 3 ax 12 ab + 4 ax

2. 8 ac? — 362

7 ac 872
3 ac? — 962

3. 15 ab3c4 – 12 abc 12 ab%c" – 15 abc

absc* - abc

32. When the Terms are Similar and have Unlike Signs. The signs + and — stand in direct opposition to each other.

If a merchant writes + before his gains, and before his losses, at the end of the year the sum of the plus numbers will denote the gains, and the sum of the minus numbers the losses. If the gains exceed the losses, the difference, which is called the algebraic sum, will be plus; but if the losses exceed the gains, the algebraic sum will be minus.

(1) A merchant in trade gained $1500 in the first quarter of the year, $3000 in the second quarter, but lost $3000 in the third quarter, and $800 in the fourth. What was the result of the year's business? 1st quarter

+ 1500
3d quarter

- 3000
2d
+ 3000 4th

800
+ 4500

- 3800 + 4500 – 3800 = + 700, or $700 gain. (2) A merchant in trade gained $1000 in the first quarter, and $2000 the second quarter. In the third quarter he lost $1500, and in the fourth quarter $1800. What was the result of the year's business? 1st quarter ..

+ 1000
3d quarter

- 1500
2d

+ 2000
4th

1800

+ 3000

- 3300

+ 3000 – 3300 = -300, or $300 loss.

(3) A merchant in the first half year gained a dollars, and lost 6 dollars. In the second half year he lost a dollars, and gained b dollars. What is the result of the year's business ?

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Hence the algebraic sum of a positive and a negative quantify is their arithmetical difference, with the sign of the greater prefixed. Add the following: (4) 8 ab

(5)

4 acha (6) – 4 a2b2c
3 ab
-8acba

+ 6 a'b'c*
- 6 ab

acb?

- 2 apbca
5 ab
- 3 acb?

0

Hence, when the terms are similar and have unlike signs, Write the similar terms in the same column.

Add the coefficients of the additive terms and also the coefficients of the subtractive terms.

Take the difference of these sums, prefix the sign of the greater, and then annex the literal part or unit.

Exercises.

What is the sum of 1. 2 a 78 -- 5 a'l8 + 7 a'b3 + 6 a'53 – 11 abs?

Having written the similar terms in the same column, we find the sum of the positive coefficients to be 15, and the sum of the negative coefficients to be – 16. Their difference is – 1: hence the sum is - a268.

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2. 3a'b + 5a’b -- 3a2b + 4a2b-6a'b -- a'b ? Ans. 2ab. 3. 12abc? 4 a’bc? + 6 abc* — 8 a'bc? + 11 a*bc? ? Ans. 17 a'bc. 4. 4 a'b - 8 a'b-9ab + 11 ab?

Ans. — 2 a’b. 5. 7 abc - abc 7 abc + 8 abc + 6 abc?? Ans. 13 abc. 6. 9cb3 5 c63 — 8 ac + 20 cb3 + 9 acé 24 c63 ? Ans. + ac.

33. To Add any Algebraic Quantities.
(1) What is the sum of 3 a, 56, and -2c?

Write the quantities thus: 3 a +56 – 2c, which indicates their sum, as the terms are dissimilar; that is, have no common unit. (2) Let it be required to find the sum of the quantities

2a? - 4ab
3a – 3 ab + b2

2 ab - 562

5 a? - 5ab 462 From the preceding examples we have, for the addition of algebraic quantities, the following rule:

Write the quantities to be added, placing similar terms in the same column, and giving to each its proper sign.

Add each column separately, and then annex the dissimilar terms with their proper signs.

Exercises. 1. Add the polynomials 3a – 26 4 ab, 5 a? + 2ab, and 3 ab - 3* 26.

The term 3 a2 being similar to 5a?, we 3 fe? – 44-2 write 8 a? for the result of the reduction of

542? +261 - M ? these two terms, at the same time slightly crossing them, as in the first term.

+3448 - 2*2 — 342 Passing then to the term -- 4. ab, which . 8 a + ab.-56? — 3 c? is similar to + 2 ab and +3 ab, the three reduce to + ab, which is placed after 8 a?, and the terms crossed like the first term. Passing then to the terms involving b?, we find their sum to be – 562, after which we write - 3c2.

NOTE. — The marks are drawn across the terms, that none of them may be overlooked or omitted.

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5. If a=

-5, b=4, c=2, x=1, what are the numerical values of the several sums above found ?

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15. Add-6+3c-d-115e + 6f - 59,

36-2c-3d-e +- 27f, 5c-8d+3f 79,
- 76 - 60+ 17d+90-5f +119,
- 36 -- 5d - 2e +6 -- 99 + h.

Ans. -86 -- 109e +37f-10g + h.

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